Question

The following table provides the expected return and the standard deviation of returns for srocks and gold. Your client is currently holding a portfolio of stocks and he is considering whether he should replace half of the stocks with gold.

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Question 1 Part a) The following table provides the expected return and the standard deviation of returns for stocks and gold. Your client is currently holding a portfolio of stocks and he is considering whether he should replace haif of the stocks with gold. Expected retun Standard deviation24%- Stocks 16% Gold 12% 28%. Calculate the portfolio expected returns and standard deviation, and discuss how you would advise your client, based on the following information: i) If the correlation coefficient between the returns on gold and stocks was 0. (10 marks) i) If the correlation coefficient between the returns on gold and stocks was 1 . (10 marks) Part b) Discuss the concept of the two-fund separation theorem (10 marks) Part c) You are comparing the performance of two investment portfolios. One averaged 18% return and the other a 15% return. However, the beta of the first portfolio was 1.5, while that of the second was 0.9. i) If the T-bill rate was 5% and the market return during the period was 15%, which portfolio would generate superior risk-adjusted abnormal return according to CAPM? (10 marks) ii) Let time 0 be the present and Pi be the future value in time 1. The time value of money suggests that the present value of P can be calculated as Po= where k is the appropriate discount rate. Use the CAPM to derive what k should be. Does your result justify the use of risk free rate of return for k when there is no uncertainty on the (10 marks) (Total 50 marks) value of P/?

Answer 1

Part a)

The expected return on your portfolio is simply a weighted average of the expected returns on the individual stocks. Therefore,

Expected portfolio return= 0.50*16+0.50*12

Expected portfolio return=14%

i) The variance for a portfolio consisting of two assets is calculated using the following formula:

= σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov_1,2

If the Correlation coefficient between the returns on gold and stock was 0, the standard deviation would be -

  = (0.50)²(0.24) ² + (0.50) ² (0.28) ² + 2(0.50)(0.50)(0.24)(0.28)(0.30) (0)

= 0.0144+0.0196+0

=0.034

Standard deviation = Sqrt(0.034) = 0.1844 OR 18.44%

ii) If the Correlation coefficient between the returns on gold and stock was 1, the standard deviation would be-

= (0.50)²(0.24) ² + (0.50) ² (0.28) ² + 2(0.50)(0.50)(0.24)(0.28)(0.30) (1)

= 0.0144+0.0196+0.01008

= 0.0441

Standard deviation = Sqrt(0.0441) =0.21 OR 21%

Advice - The portfolio risk is now less(18%/21%) than risk of investing in stocks(24%) alone, so if the investor is concerned about risk he should consider to invest in gold alongwith stocks.

Part b)

The two-fund separation theorem is a central result in modern portfolio theory pioneered by Markowitz. It tells us that the risk-return pair of any admissible or feasible portfolio cannot lie above the capital market line (CML) in the risk-return space, obtained by combining the risk-free asset and the portfolio that maximizes the Sharpe ration (SR).

In this theory,Portfolio choice is separated into two stages:

- Find the efficient portfolio of risky assets;

- Find the optimum fraction to invest in the efficient portfolio of risky assets and the risk-free asset.

The role of risk aversion is confined to the second stage and plays no role in the first stage.

The two-fund separation theorem tells us that an investor with quadratic utility can separate her asset allocation decision into two steps:

First, find the tangency portfolio (TP), i.e., the portfolio of risky assets that maximizes the Sharpe ratio (SR); and then, decide on the mix of the TP and the risk-free asset, depending on the investor’s attitude toward risk.

Part c)

i)Risk adjusted Abnormal Rate of Return according to CAPM , would be calculated as follows-

r = Rf + beta * (Rm - Rf ) + Risk adjusted abnormal rate of return

1st portfolio = Return =18%, Beta = 1.5

18= 5+ (15-5)1.5+ Risk adjusted Abnormal Rate of return

18-20 = Risk adjusted Abnormal Rate of Return

Risk adjusted Abnormal Rate of Return = -2 %

2nd portfolio = Return =15%, Beta = 0.9

15= 5+(15-5)0.9+Risk adjusted Abnormal Rate of Return

1% = Risk adjusted Abnormal Rate of Return

Portfolio 2 would generate superior Risk adjusted Abnormal Rate of Return according to CAPM

ii) The CAPM formula for using appropriate discount rate is

Ke = Rf + beta * (Rm - Rf )

In the absence of systematic risk or no uncertainty, the discounted rate will be the risk free rate as the beta will be

zero of a portfolio with no uncertainty. Thus risk free rate can be used as k in when there is no uncertainty on the the

value of P1